I understand that with continuous and strictly increasing utility function we can find Pareto efficient allocations by looking at the allocation that satisfies $MRS_1 = MRS_2$. However, I was not sure how we can find Pareto Efficient Allocations for discontinuous utilities (or one continuous and one discontinuous utility).
For instance, given exchange economy with one unit of initial endowment of the good $x$ and $y$, if we have person A's utility function as $$u_A(x,y) = x^\alpha y^{1-\alpha}$$ and person B's preference is represented by lexicographic: given $(x,y)$ and $(x',y')$, B prefers the former if $$x>x'$$ or $$x = x', y>y'$$ In this case, we can't calculate $MRS_B$. Does anyone have any suggestion how can we find the Pareto Efficient Allocation and Core Allocation? Thank you very much for your help in advance!